Fitting sets in U-groups Olivier Frécon To cite this version: Olivier Frécon. Fitting sets in U-groups. 2013. hal-00912367 HAL Id: hal-00912367 https://hal.archives-ouvertes.fr/hal-00912367 Preprint submitted on 2 Dec 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FITTING SETS IN U-GROUPS OLIVIER FRECON´ Abstract. We consider different natural definitions for Fitting sets, and we prove the existence and conjugacy of the associated injectors in the largest possible classes of locally finite groups. A Fitting set X of a finite group G is a non-empty set of subgroups which is closed under taking normal products, taking subnormal subgroups and conjugation in G. This concept of Anderson generalized the work of Fischer, Gasch¨utz and Hartley about Fitting classes [2, p.535–537], where a Fitting class is a class of finite groups closed under taking normal products and taking subnormal subgroups. The theory of Fitting sets concern the study of injectors: for a collection X of finite groups, a subgroup V of a finite group G is an X-injector of G if V ∩ A is a maximal X-subgroup of A for every subnormal subgroup A. The basic theorem provides the existence and conjugacy of X-injectors in any finite soluble group G for each Fitting set X [2, Theorem 2.9 p.539]. In this paper, we consider the natural generalizations of Fitting sets and injectors to locally finite groups. For each definition of a Fitting set, we prove the existence and conjugacy of associated injectors in a class of locally finite groups. It is shown in [3] that our classes of locally finite groups are the largest in which the results about injectors in finite soluble groups hold. Note that, thanks to our new approach, we generalize in the last section the theorem of Hartley and Tomkinson about the existence and conjugacy of injectors in U-groups [6] from Fitting classes to Fitting sets. 1. Normal Fitting sets In this section, we analyze the injectors associated to normal Fitting sets. Notation 1.1. Let X be a collection of groups. For each group G, we denote by GXn the join of its normal X-subgroups. Definition 1.2. Let G be a locally finite group. A nonempty set X of subgroups of G is a normal Fitting set of G if: (NF 1) every normal subgroup of an X-group belongs to X; (NF 2) when H is a subgroup of G then HXn is an X-subgroup of H; (NF 3) every conjugate of an X-group is an X-group. Definition 1.3. Let G be a locally finite group. If X is a set of subgroups of G, a subgroup V of G is an X-n-injector of G if, for every normal subgroup A of G, V ∩ A is a maximal X-subgroup of A. Date: November 10, 2013. 1991 Mathematics Subject Classification. Primary 20F17; Secondary 20D10. Key words and phrases. locally finite groups, Fitting sets, Injectors. 1 2 OLIVIER FRECON´ The main result of this section is the following. Theorem 1.4. Let G be a subsoluble U-group, and X a normal Fitting set of G. Then G has exactly one conjugacy class of X-n-injectors. It is likely that the class of subsoluble U-groups is the largest class of groups satisfying this result. A partial answer is obtained in [3]. We recall that a group is called a Baer group (resp. a Gruenberg group) if every cyclic subgroup is subnormal (resp. ascendant). A group G is said to be subsoluble (resp. an SN ∗-group) if it possesses an ascending series with each term subnormal (resp. ascendant) in G and each factor abelian. In any group G, there is a Baer radical β(G) which is a Baer subgroup containing all the subnormal Baer subgroups of G, and a Gruenberg radical γ(G) which is a Gruenberg subgroup containing all the ascendant Gruenberg subgroups of G. We refer to [7, §2.3] for more details. As in [1, p.15], we denote by ρ(G) the Hirsch-Plotkin radical of any group G, and ρ0(G) = 1 and ρn+1(G)/ρn(G)= ρ(G/ρn(G)) for each integer n. We recall that the class U was introduced as below in [4], and that by an Hartley’s Theorem [1, Theorem 4.4.7 p.163], the first condition is redundant. Definition 1.5. The class U is the largest subgroup-closed class of locally finite groups satisfying the conditions: (U1) if G ∈ U then G = ρn(G) for an integer n; (U2) if G ∈ U and π is any set of primes, then the Sylow π-subgroups of G are conjugate in G. Fact 1.6. [5, Theorem E] If G is a U-group, then G/ρ(G) is (2-step soluble)-by-finite. In particular G/ρ(G) is soluble. Moreover, G/ρ(G) is hyperfinite. We need to introduce normal∗ Fitting sets as the sets X of subgroups of a locally finite group G satisfying the conditions (NF 1), (NF 3) and ∗ (NF 2 ) if H is a subgroup of G such that H/HXn is locally nilpotent, then HXn is a maximal X-subgroup of H. Any normal∗ Fitting set of a locally finite group G is a normal Fitting set too. By the result below, the converse is true in subsoluble U-groups. Proposition 1.7. If X is a normal Fitting set of a subsoluble U-group G, then X is a normal∗ Fitting set. Proof. Let H be a subgroup of G such that H/HXn is locally nilpotent. We define δ0(H) = HXn, δi+1(H)/δi(H) = β(H/δi(H)) for every ordinal i, and δµ(H) = ∪i<µδi(H) for every limit ordinal µ. Since G is subsoluble, then H is subsoluble too, and there is an ordinal α such that δα(H)= H. Suppose toward a contradiction that HXn is not a maximal X-subgroup of H. Since it is an X-subgroup by (NF 2), there is a smallest ordinal γ such that HXn is strictly contained in an X-subgroup K of δγ (H). By (NF 1), we have K ∩ δi(H) = HXn for all i<γ, so there is an ordinal ν such that γ = ν + 1. Now K/HXn ≃ Kδν (H)/δν (H) is a Baer group. Hence, if g is an element of K, then hgiHXn is subnormal in K, and hgiHXn is an X-group by (NF 1). By the minimality of γ, it is a maximal X-subgroup of hgiδi(H) for all i<γ. For any element b of δν (H), the local nilpotence of H/HXn implies the nilpotence of hg,biHXn/HXn. Therefore hgiHXn is a subnormal X-subgroup of hg,biHXn. By FITTING SETS IN U-GROUPS 3 the maximality of hgiHXn in hgiδν (H), we have hgiHXn = (hg,biHXn)Xn and b normalizes hgiHXn, so hgiHXn =(hgiδν (H))Xn. Since δγ (H)/δν (H) is a Baer group, then hgiδν (H) is subnormal in δγ (H). Hence hgiHXn is a subnormal X-subgroup of δγ (H) and we have hgiHXn ≤ (δγ (H))Xn. As δγ (H) is a normal subgroup of H, we obtain hgiHXn ≤ HXn, contradicting the choice of g. This proves the proposition. For this section, we fix a normal∗ Fitting set F of a fixed U-group G. Lemma 1.8. Let A a subgroup of G containing G′. If A possesses a maximal F- subgroup W , then every F-subgroup of G containing W is contained in a maximal F-subgroup of G. Proof. Let (Vi)i<α be an increasing sequence of F-subgroups of G containing W , for an ordinal α. For each i<α, since A is normal in G, Vi ∩ A is an F-subgroup of A containing W and we have Vi ∩ A = W by the maximality of W . As A contains ′ ′ G , then W contains V where V = ∪i<αVi. So Vi is normal in V for all i<α. We ∗ obtain V = VFn and V is an F-group by (NF 2 ). A Carter subgroup of a group H is a locally nilpotent and self-serializing subgroup of H. The main result concerning these subgroups is the following: Fact 1.9. [4, Theorem 5.4] Let H be a U-group, and A a normal subgroup of H. Then H has a unique conjugacy class of Carter subgroups. Moreover, if C is a Carter subgroup of H, then CA/A is a Carter subgroup of H/A and each Carter subgroup of H/A has this form. Lemma 1.10. Let A be a subgroup of G containing G′. If A has a maximal F- subgroup W , then the maximal F-subgroups of G containing W are conjugate. Proof. By Lemma 1.8, G has a maximal F-subgroup V containing W , and W = V ∩ A is normal in V . Let U = NG(W ) and N/W = NU/W (V/W ). We show that there exists a Carter subgroup C/W of U/W such that V = CFn.